Tracking Volatility

We propose an adaptive algorithm for tracking historical volatility. The algorithm borrows ideas from nonparametric statistics. In particular, we assume that the volatility is a several times differentiable function with a bounded highest derivative. We propose an adaptive algorithm with a Kalman filter structure, which guarantees the same asymptotics (well known from statistical inference) with respect to the sample size n, n → ∞. The tuning procedure for this filter is simpler than for a GARCH filter.     

Nonlinear filters for linear models (a robust approach)

We consider the altering problem for linear models where the driving noises may be quite general, nonwhite and non-Gaussian, and where the observation noise may only be known to belong to a finite family of possible disturbances. Using diffusion approximation methods, we show that a certain nonlinear filter minimizes the asymptotic filter variance. This nonlinear filter is obtained by choosing at each moment, on the basis of the observations, one of a finite number of Kalman-type filters driven by a suitable nonlinear transformation of the “innovations”. As a byproduct we obtain also the asymptotic identification of the a priori unknown observation noise disturbance. By yielding an asymptotically efficient filter in face of an unknown observation noise, our approach may also be viewed as a robust approach to filtering for linear models     

Markovianity of a subset of components of a Markov process

We consider the necessary and sufficient conditions for a group of the components of a stationary vector Gaussian Markov process to possess Markov property. The representation by a linear Itô stochastic differential equation is also given.